Rung 30: security-constrained — a meshed triangle of lines and two transformers, one extendable, that must carry their flow within their rating after any one of three outages¶
One rung of the PyPSA corpus: the file pypsa.yaml projected onto what this network builds, attached to that network, and held to what PyPSA solves it to.
✔ Verified against pypsa master at 51986084 — objective 22113.333333 on both sides; structure ✔ 26 constraints · 6 variables, name for name; size ✔ 240 rows · ✔ 46 columns · ✔ 412 nonzeros; duals ✔ 240 rows; model for model: 32 blocks equal, 0 documented splits, 1 recorded deviations.
Rows and columns, PyPSA against specsolve, name for name
| row | PyPSA | specsolve |
|---|---|---|
Bus-nodal_balance |
20 | 20 |
Generator-fix-p-lower |
20 | 20 |
Generator-fix-p-upper |
20 | 20 |
Kirchhoff-Voltage-Law |
8 | 8 |
Line-ext-s-lower |
4 | 4 |
Line-ext-s-lower-security-for-{c}-outage-in-sub-network-{n} |
12 | 12 |
Line-ext-s-upper |
4 | 4 |
Line-ext-s-upper-security-for-{c}-outage-in-sub-network-{n} |
12 | 12 |
Line-ext-s_nom-lower |
1 | 1 |
Line-ext-s_nom-upper |
1 | 1 |
Line-fix-s-lower |
8 | 8 |
Line-fix-s-lower-security-for-{c}-outage-in-sub-network-{n} |
24 | 24 |
Line-fix-s-upper |
8 | 8 |
Line-fix-s-upper-security-for-{c}-outage-in-sub-network-{n} |
24 | 24 |
Link-fix-p-lower |
4 | 4 |
Link-fix-p-upper |
4 | 4 |
Transformer-ext-s-lower |
4 | 4 |
Transformer-ext-s-lower-security-for-{c}-outage-in-sub-network-{n} |
12 | 12 |
Transformer-ext-s-upper |
4 | 4 |
Transformer-ext-s-upper-security-for-{c}-outage-in-sub-network-{n} |
12 | 12 |
Transformer-ext-s_nom-lower |
1 | 1 |
Transformer-ext-s_nom-upper |
1 | 1 |
Transformer-fix-s-lower |
4 | 4 |
Transformer-fix-s-lower-security-for-{c}-outage-in-sub-network-{n} |
12 | 12 |
Transformer-fix-s-upper |
4 | 4 |
Transformer-fix-s-upper-security-for-{c}-outage-in-sub-network-{n} |
12 | 12 |
| column | PyPSA | specsolve |
|---|---|---|
Generator-p |
20 | 20 |
Line-s |
12 | 12 |
Line-s_nom |
1 | 1 |
Link-p |
4 | 4 |
Transformer-s |
8 | 8 |
Transformer-s_nom |
1 | 1 |
The model¶
The same model, as math
A plain n.optimize(), and its multi-period and stochastic classes, in one file. Every second-stage quantity spans a scenario (a future dispatch is chosen in) and every asset stands in the investment periods its build year and lifetime span. A parameter spans scenario exactly when PyPSA reads it per scenario. Capacity is chosen once, before the future is known, and paid once per active period at its cost in expectation over the scenarios; operation is the expectation over the scenarios' weights, with a share priced at the tail through the CVaR rows, which stand only where a risk preference is set. A plain run feeds one scenario, one period, all-active masks and unit weights, and the model collapses to the standard one. A security-constrained run copies each branch flow limit once per outage in an outage set that a plain run leaves empty. Which snapshots an asset is active in, a scenario's weight, and the outage factors are data prep.
Sets¶
| Symbol | Meaning |
|---|---|
| \(\Xi\) | index \(\xi\) — scenario — the futures dispatch is chosen in, each with a weight |
| \(\mathcal{T}\) | index \(t\) — snapshot with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — dispatch periods |
| \(\mathcal{N}\) | index \(n\) — bus with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N},\ \mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N},\ \mathrm{Process\_output\_bus}: \mathcal{R} \to \mathcal{N},\ \mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N},\ \mathrm{Line\_bus0}: \mathcal{K} \to \mathcal{N},\ \mathrm{Line\_bus1}: \mathcal{K} \to \mathcal{N},\ \mathrm{Transformer\_bus0}: \mathcal{M} \to \mathcal{N},\ \mathrm{Transformer\_bus1}: \mathcal{M} \to \mathcal{N}\) — network nodes |
| \(\mathcal{G}\) | index \(g\) — generator with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N}\) — generating units, each on one bus |
| \(\mathcal{L}\) | index \(l\) — link with \(\mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L}\) — controllable connections, each from one bus to the buses it delivers to |
| \(\mathcal{O}\) | index \(o\) — link_output with \(\mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N}\) — a link's output ports, one label per port a link declares — PyPSA's bus1, bus2, … columns read long, so a link of any number of output ports is one term in the balance, data prep |
| \(\mathcal{J}\) | index \(j\) — process with \(\mathrm{Process\_output\_process}: \mathcal{R} \to \mathcal{J}\) — generalized multi-port converters, each with an internal power that every port draws or delivers at its own rate |
| \(\mathcal{R}\) | index \(r\) — process_output with \(\mathrm{Process\_output\_process}: \mathcal{R} \to \mathcal{J},\ \mathrm{Process\_output\_bus}: \mathcal{R} \to \mathcal{N}\) — a process's ports, one label per port a process declares — PyPSA's bus0, bus1, … each carry a signed rate, so a process of any number of ports is one term in the balance, data prep |
| \(\mathcal{D}\) | index \(d\) — load with \(\mathrm{Load\_bus}: \mathcal{D} \to \mathcal{N}\) — demands, each on one bus |
| \(\mathcal{K}\) | index \(k\) — line with \(\mathrm{Line\_bus0}: \mathcal{K} \to \mathcal{N},\ \mathrm{Line\_bus1}: \mathcal{K} \to \mathcal{N},\ \mathrm{Outage\_line}: \mathcal{K}^{\mathrm{out}} \to \mathcal{K}\), Outage_line is absent where the data has no row — passive branches, each between two buses, their flow set by impedance |
| \(\mathcal{M}\) | index \(m\) — transformer with \(\mathrm{Transformer\_bus0}: \mathcal{M} \to \mathcal{N},\ \mathrm{Transformer\_bus1}: \mathcal{M} \to \mathcal{N},\ \mathrm{Outage\_transformer}: \mathcal{K}^{\mathrm{out}} \to \mathcal{M}\), Outage_transformer is absent where the data has no row — passive branches between two buses, their flow set by impedance and tap ratio, with a phase shift fixed or optimised |
| \(\mathcal{C}\) | index \(c\) — cycle — independent cycles of the passive network graph — the cycle basis, data prep. Each period has its own basis, of the branches that stand in it; a label is a position in that period's basis, so one label names a different cycle in another period |
| \(\mathcal{K}^{\mathrm{out}}\) | index \(\kappa\) — outage with \(\mathrm{Outage\_line}: \mathcal{K}^{\mathrm{out}} \to \mathcal{K},\ \mathrm{Outage\_transformer}: \mathcal{K}^{\mathrm{out}} \to \mathcal{M}\), Outage_line is absent where the data has no row, Outage_transformer is absent where the data has no row — the passive branches a security-constrained run takes out one at a time — PyPSA's branch_outages, each a line or a transformer; none on a plain run |
| \(\mathcal{Y}\) | index \(y\) — period with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — investment periods — PyPSA's investment_periods |
Parameters¶
| Symbol | Meaning |
|---|---|
| \(\mathrm{w}\) | snapshot_weightings_objective over \(\mathcal{T}\) — PyPSA's snapshot_weightings.objective — hours a snapshot stands for in the cost |
| \(\mathrm{p}^{\mathrm{nom}}\) | Generator_p_nom over \(\Xi \times \mathcal{G}\) — nominal power |
| \(\mathrm{ext}\) | Generator_p_nom_extendable over \(\mathcal{G}\) — whether the nominal power is a decision |
| \(\underline{\mathrm{p}}\) | Generator_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — least output, per unit of nominal power |
| \(\overline{\mathrm{p}}\) | Generator_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — most output, per unit of nominal power — an availability profile |
| \(\mathrm{c}\) | Generator_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of one unit of output |
| \(\mathrm{c}^{(2)}\) | Generator_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of the square of one unit of output |
| \(\mathrm{sgn}\) | Generator_sign over \(\mathcal{G}\) — the sign output enters its bus's balance with — PyPSA's sign, 1 unless given, -1 for a unit that draws power. PyPSA refuses one that differs by scenario (constants.py:43) |
| \(\mathrm{com}\) | Generator_committable over \(\mathcal{G}\) — whether output is gated by an on/off status decision |
| \(\mathrm{f}^{\mathrm{nom}}\) | Link_p_nom over \(\Xi \times \mathcal{L}\) — nominal power |
| \(\mathrm{ext}^{f}\) | Link_p_nom_extendable over \(\mathcal{L}\) — whether the nominal power is a decision |
| \(\underline{\mathrm{f}}\) | Link_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — least flow, per unit of nominal power — negative for a link that carries both ways |
| \(\overline{\mathrm{f}}\) | Link_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — most flow, per unit of nominal power |
| \(\eta\) | Link_efficiency over \(\Xi \times \mathcal{T} \times \mathcal{O}\) — share of the flow that arrives at an output port, PyPSA's efficiency, efficiency2, … read long — negative where that port consumes rather than delivers. Read at the snapshot the flow arrives, so a delayed port delivers at its arrival snapshot's efficiency (constraints.py:1498) |
| \(\mathrm{d}^{f}\) | Link_output_delay over \(\mathcal{O}\) — snapshots a port's delivery lags its link's flow — PyPSA's delay, delay2, … read long, divided by the snapshot_weightings.generators value and rounded up in data prep; zero for a port that delivers at once. A port has the same delay in every scenario, because PyPSA refuses a delay that differs by scenario (constants.py:52) |
| \(\mathrm{cyc}^{f}\) | Link_output_cyclic_delay over \(\mathcal{O}\) — whether a delayed port's flow wraps from the end of its investment period — PyPSA's cyclic_delay, cyclic_delay2, …; where it does not, the flow still in transit at each period's first snapshots is lost. The flag is the same in every scenario, as the delay is |
| \(\mathrm{c}^{f}\) | Link_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — cost of one unit of flow |
| \(\mathrm{c}^{f,(2)}\) | Link_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — cost of the square of one unit of flow |
| \(\mathrm{com}^{f}\) | Link_committable over \(\mathcal{L}\) — whether flow is gated by an on/off status decision |
| \(\alpha\) | Process_rate over \(\Xi \times \mathcal{T} \times \mathcal{R}\) — the energy a port draws or delivers per unit of internal power, PyPSA's rate0, rate1, … read long — negative where the port withdraws, positive where it injects; a link is a process whose bus0 rate is minus one and whose output rates are its efficiencies. Read at the snapshot the transfer arrives, so a delayed port transfers at its arrival snapshot's rate (constraints.py:1498) |
| \(\mathrm{d}^{z}\) | Process_output_delay over \(\mathcal{R}\) — snapshots a port's transfer lags its process's internal power — PyPSA's delay0, delay1, … read long, divided by the snapshot_weightings.generators value and rounded up in data prep; zero for a port that transfers at once. A port has the same delay in every scenario, as a link's port has |
| \(\mathrm{cyc}^{z}\) | Process_output_cyclic_delay over \(\mathcal{R}\) — whether a delayed port's transfer wraps from the end of its investment period — PyPSA's cyclic_delay0, cyclic_delay1, …; where it does not, the energy still in transit at each period's first snapshots is lost. The flag is the same in every scenario, as the delay is |
| \(\mathrm{load}\) | Load_p_set over \(\Xi \times \mathcal{T} \times \mathcal{D}\) — demand |
| \(\mathrm{sgn}^{\mathrm{load}}\) | Load_sign over \(\mathcal{D}\) — the sign a load's demand enters its bus's balance with — PyPSA's sign, -1 unless given, 1 for a load that feeds its bus. PyPSA refuses one that differs by scenario (constants.py:43) |
| \(\mathrm{on}^{\mathrm{load}}\) | Load_active over \(\mathcal{D}\) — whether a load stands in the model — PyPSA's active. A load has no build year and no lifetime, so the flag holds in every snapshot. PyPSA refuses one that differs by scenario (constants.py:51) |
| \(\pi\) | scenario_weight over \(\Xi\) — PyPSA's scenario_weightings.weight — the probability of a future |
| \(\mathrm{w}^{y}\) | period_weight_objective over \(\mathcal{Y}\) — PyPSA's investment_period_weightings.objective — what a period's cost weighs; PyPSA reads it only under multi_investment_periods, so data prep feeds one otherwise, whatever the column holds (optimize.py:205-207, :264-266) |
| \(\mathrm{on}\) | Generator_active over \(\mathcal{T} \times \mathcal{G}\) — whether a generator stands in a snapshot's period — PyPSA's active, from build year and lifetime, data prep |
| \(\mathrm{on}^{f}\) | Link_active over \(\mathcal{T} \times \mathcal{L}\) — whether a link stands in a snapshot's period — PyPSA's active, data prep |
| \(\mathrm{on}^{s}\) | Line_active over \(\mathcal{T} \times \mathcal{K}\) — whether a line stands in a snapshot's period — PyPSA's active, data prep |
| \(\mathrm{on}^{z}\) | Process_active over \(\mathcal{T} \times \mathcal{J}\) — whether a process stands in a snapshot's period — PyPSA's active, data prep |
| \(\mathrm{on}^{\sigma}\) | Transformer_active over \(\mathcal{T} \times \mathcal{M}\) — whether a transformer stands in a snapshot's period — PyPSA's active, data prep |
| \(\mathrm{act}^{s}\) | Line_active_asset over \(\mathcal{K}\) — whether a line is one of PyPSA's active_assets — its active flag alone, without build year and lifetime, so a unit that stands in no snapshot of the horizon still counts. PyPSA refuses one that differs by scenario (constants.py:51) |
| \(\mathrm{act}^{\sigma}\) | Transformer_active_asset over \(\mathcal{M}\) — whether a transformer is one of PyPSA's active_assets — its active flag alone, without build year and lifetime, so a unit that stands in no snapshot of the horizon still counts. PyPSA refuses one that differs by scenario (constants.py:51) |
| \(\mathrm{W}^{s}\) | Line_capital_weight over \(\mathcal{K}\) — the sum of period weights a line stands in — PyPSA's active * period_weighting, summed, data prep |
| \(\mathrm{W}^{\sigma}\) | Transformer_capital_weight over \(\mathcal{M}\) — the sum of period weights a transformer stands in — PyPSA's active * period_weighting, summed, data prep |
| \(\mathrm{s}^{\mathrm{nom}}\) | Line_s_nom over \(\Xi \times \mathcal{K}\) — nominal apparent power |
| \(\mathrm{ext}^{s}\) | Line_s_nom_extendable over \(\mathcal{K}\) — whether the nominal apparent power is a decision |
| \(\overline{\mathrm{s}}\) | Line_s_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{K}\) — most flow either way, per unit of nominal apparent power |
| \(\underline{\mathrm{s}}^{\mathrm{nom}}\) | Line_s_nom_min over \(\Xi \times \mathcal{K}\) — least nominal apparent power an extendable line may be built at |
| \(\overline{\mathrm{s}}^{\mathrm{nom}}\) | Line_s_nom_max over \(\Xi \times \mathcal{K}\) — most nominal apparent power an extendable line may be built at |
| \(\mathrm{c}^{\mathrm{cap},s}\) | Line_capital_cost over \(\Xi \times \mathcal{K}\) — cost of one unit of nominal apparent power for the modelled horizon — PyPSA's periodized_cost: overnight_cost as an annuity over lifetime at discount_rate, times nyears, where it is given, and capital_cost where it is not, plus fom_cost (components.py:1126-1147, costs.py:102-203), data prep |
| \(\mathrm{x}\) | Line_cycle_weight over \(\mathcal{Y} \times \mathcal{K} \times \mathcal{C}\), neutral where the data has no row — the line's series impedance, signed by its orientation in the cycle — the cycle basis, data prep; a line in no cycle has no row. PyPSA builds the cycle basis from the first scenario only (networks.py:1430-1437) |
| \(\beta\) | Line_BODF over \(\mathcal{Y} \times \mathcal{K} \times \mathcal{K}^{\mathrm{out}}\), neutral where the data has no row — the share of an outaged branch's flow a line takes on when that branch goes out — PyPSA's BODF, from the PTDF of the sub-network the period's active branches form, data prep; a row only where the line and the outage are active in the period and share a sub-network, -1 at the outaged line itself. The factors are the same in every scenario, because PyPSA takes those of the first scenario (abstract.py:534) |
| \(\sigma^{\mathrm{nom}}\) | Transformer_s_nom over \(\Xi \times \mathcal{M}\) — nominal apparent power |
| \(\mathrm{ext}^{\sigma}\) | Transformer_s_nom_extendable over \(\mathcal{M}\) — whether the nominal apparent power is a decision |
| \(\overline{\sigma}\) | Transformer_s_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{M}\) — most flow either way, per unit of nominal apparent power |
| \(\underline{\sigma}^{\mathrm{nom}}\) | Transformer_s_nom_min over \(\Xi \times \mathcal{M}\) — least nominal apparent power an extendable transformer may be built at |
| \(\overline{\sigma}^{\mathrm{nom}}\) | Transformer_s_nom_max over \(\Xi \times \mathcal{M}\) — most nominal apparent power an extendable transformer may be built at |
| \(\mathrm{c}^{\mathrm{cap},\sigma}\) | Transformer_capital_cost over \(\Xi \times \mathcal{M}\) — cost of one unit of nominal apparent power for the modelled horizon — PyPSA's periodized_cost: overnight_cost as an annuity over lifetime at discount_rate, times nyears, where it is given, and capital_cost where it is not, plus fom_cost (components.py:1126-1147, costs.py:102-203), data prep |
| \(\mathrm{x}^{\sigma}\) | Transformer_cycle_weight over \(\mathcal{Y} \times \mathcal{M} \times \mathcal{C}\), neutral where the data has no row — the transformer's effective series reactance, x times its tap ratio, signed by its orientation in the cycle — PyPSA's x_pu_eff, the cycle basis, data prep; a transformer in no cycle has no row. From the first scenario only, as a line's |
| \(\beta^{\sigma}\) | Transformer_BODF over \(\mathcal{Y} \times \mathcal{M} \times \mathcal{K}^{\mathrm{out}}\), neutral where the data has no row — the share of an outaged branch's flow a transformer takes on when that branch goes out, as a line's; a row only where the transformer and the outage are active in the period and share a sub-network |
Variables¶
| Symbol | Meaning |
|---|---|
| \(p\) | Generator_p over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-p — output of a generator in a snapshot |
| \(f\) | Link_p over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — Link-p — PyPSA's p0, the flow measured at the Link_bus0 end: a positive value withdraws there and injects at every bus the link's output ports deliver to |
| \(s\) | Line_s over \(\Xi \times \mathcal{T} \times \mathcal{K}\) — Line-s — PyPSA's p0, the flow measured at the Line_bus0 end: a positive value withdraws there and injects at Line_bus1, lossless |
| \(\sigma\) | Transformer_s over \(\Xi \times \mathcal{T} \times \mathcal{M}\) — Transformer-s — PyPSA's p0, the flow measured at the Transformer_bus0 end: a positive value withdraws there and injects at Transformer_bus1, lossless |
| \(S\) | Line_s_nom_ext over \(\mathcal{K}\) — Line-s_nom — nominal apparent power where it is a decision, for a line that is active; the parameter of the same PyPSA name carries the fixed regime |
| \(\Sigma\) | Transformer_s_nom_ext over \(\mathcal{M}\) — Transformer-s_nom — nominal apparent power where it is a decision, for a transformer that is active; the parameter of the same PyPSA name carries the fixed regime |
| \(z\) | Process_p over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — Process-p — PyPSA's internal power p: a positive value drives every port at its own rate, withdrawing where the rate is negative and injecting where it is positive |
Definitions¶
| Symbol | Meaning |
|---|---|
| \(\mathit{Link\_output\_arrival}\) | Link_output_arrival over \(\Xi \times \mathcal{T} \times \mathcal{O}\) — what a link delivers to an output port at a snapshot — its flow delayed by the port's delay within its investment period, times the port's efficiency at the snapshot the flow arrives; where the port is cyclic_delay the delayed flow wraps from the period's end, and where it is not the flow still in transit at the period's first snapshots is lost. A port that does not delay (delay zero) delivers its flow unshifted, cyclic or not |
| \(\mathit{Process\_output\_arrival}\) | Process_output_arrival over \(\Xi \times \mathcal{T} \times \mathcal{R}\) — what a process transfers at a port at a snapshot — its internal power delayed by the port's delay within its investment period, times the port's rate at the snapshot the transfer arrives; where the port is cyclic_delay the delayed transfer wraps from the period's end, and where it is not the energy still in transit at the period's first snapshots is lost. A port that does not delay (delay zero) transfers at once, cyclic or not |
| \(\mathit{Generator\_opex}\) | Generator_opex over \(\Xi\) |
| \(\mathit{Link\_opex}\) | Link_opex over \(\Xi\) |
| \(\mathit{scenario\_opex}\) | scenario_opex over \(\Xi\) — what a future costs to run — every operating term, weighted by the snapshot's hours and its period, before the scenario's own weight; a start and a stop cost what they cost, unweighted, as PyPSA adds them (optimize.py:415-432) |
| \(\mathit{Line\_capex}\) | Line_capex (scalar) |
| \(\mathit{Transformer\_capex}\) | Transformer_capex (scalar) |
| \(\mathit{risk\_weighted\_opex}\) | risk_weighted_opex (scalar) |
| \(\mathit{total\_cost}\) | total_cost (scalar) — what the system costs — capacity once per active period at its expected cost over the scenarios, operation in expectation over the scenarios, and a share of it at the tail |
| \(\mathrm{Load\_demand}\) | Load_demand over \(\Xi \times \mathcal{T} \times \mathcal{D}\) — what a load draws from its bus's balance — its demand times its sign where it is active, nothing where it is not, since PyPSA drops an inactive load from the balance (constraints.py:1513-1514) |
| \(\mathit{Line\_s\_monitored}\) | Line_s_monitored over \(\Xi \times \mathcal{T} \times \mathcal{K}\) — the flow a line's post-contingency rows read — its flow where it stands, nothing where it does not, since PyPSA builds those rows for every branch of the sub-network in every snapshot |
| \(\mathit{Transformer\_s\_monitored}\) | Transformer_s_monitored over \(\Xi \times \mathcal{T} \times \mathcal{M}\) — the flow a transformer's post-contingency rows read, as a line's |
| \(\mathit{Outage\_s}\) | Outage_s over \(\Xi \times \mathcal{T} \times \mathcal{K}^{\mathrm{out}}\) — the flow an outage takes off its branch — the outaged line's or transformer's flow before it goes out |
| \(\mathit{Generator\_injection}\) | Generator_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\) |
| \(\mathit{Line\_injection}\) | Line_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\) |
| \(\mathit{Link\_injection}\) | Link_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\) |
| \(\mathrm{Load\_injection}\) | Load_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\) |
| \(\mathit{Process\_injection}\) | Process_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\) |
| \(\mathit{Transformer\_injection}\) | Transformer_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\) |
| \(\mathit{Bus\_injection}\) | Bus_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\) — what every component puts into a bus, less what it takes out of it; PyPSA writes each term into the balance, and a load on its right-hand side |
| \(\mathit{Line\_angle\_sum}\) | Line_angle_sum over \(\mathcal{C} \times \Xi \times \mathcal{T}\) |
| \(\mathit{Transformer\_angle\_sum}\) | Transformer_angle_sum over \(\mathcal{C} \times \Xi \times \mathcal{T}\) |
| \(\mathit{Cycle\_angle\_sum}\) | Cycle_angle_sum over \(\Xi \times \mathcal{T} \times \mathcal{C}\) — the voltage angle differences around a cycle: every branch flow times its cycle weight, and every transformer phase shift |
Notation¶
| Symbol | Meaning |
|---|---|
| \(t \ominus k\) | edge='wrap' — \(t-k\), cyclic |
| \(t \boxminus_{v} k\) | edge=v — \(t-k\), and \(v\) past the edge |
| \(t \ominus^{\mathrm{relation}(t)} k\) | within=relation[...] — \(t-k\) inside the group of \(t\) |
Variable domains¶
Generator_p
\[
p_{\xi,t,g} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{on}_{t,g}
\]
Link_p
\[
f_{\xi,t,l} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{on}^{f}_{t,l}
\]
Line_s
\[
s_{\xi,t,k} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K} \,:\, \mathrm{on}^{s}_{t,k}
\]
Transformer_s
\[
\sigma_{\xi,t,m} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M} \,:\, \mathrm{on}^{\sigma}_{t,m}
\]
Line_s_nom_ext
\[
S_{k} \in \mathbb{R} \qquad \forall\, k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{act}^{s}_{k}
\]
Transformer_s_nom_ext
\[
\Sigma_{m} \in \mathbb{R} \qquad \forall\, m \in \mathcal{M} \,:\, \mathrm{ext}^{\sigma}_{m} \wedge \mathrm{act}^{\sigma}_{m}
\]
Process_p
\[
z_{\xi,t,j} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{on}^{z}_{t,j}
\]
Definitions¶
Link_output_arrival
\[
\mathit{Link\_output\_arrival}_{\xi,t,o} = \begin{cases} f_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{if } \mathrm{cyc}^{f}_{o} \\ f_{\xi,t \boxminus_{0}^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ o \in \mathcal{O}
\]
Process_output_arrival
\[
\mathit{Process\_output\_arrival}_{\xi,t,r} = \begin{cases} z_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{z},\mathrm{Process\_output\_process}(r)} \cdot \alpha_{\xi,t,r} & \text{if } \mathrm{cyc}^{z}_{r} \\ z_{\xi,t \boxminus_{0}^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{z},\mathrm{Process\_output\_process}(r)} \cdot \alpha_{\xi,t,r} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ r \in \mathcal{R}
\]
Generator_opex
\[
\mathit{Generator\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{\xi,t,g} \cdot \mathrm{c}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{\xi,t,g} \cdot p_{\xi,t,g} \cdot \mathrm{c}^{(2)}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
Link_opex
\[
\mathit{Link\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot \mathrm{c}^{f}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot f_{\xi,t,l} \cdot \mathrm{c}^{f,(2)}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
scenario_opex
\[
\mathit{scenario\_opex}_{\xi} = \mathit{Generator\_opex}_{\xi} + \mathit{Link\_opex}_{\xi} \qquad \forall\, \xi \in \Xi
\]
Line_capex
\[
\mathit{Line\_capex} = \sum_{\xi \in \Xi,\ k \in \mathcal{K}} \pi_{\xi} \cdot S_{k} \cdot \mathrm{c}^{\mathrm{cap},s}_{\xi,k} \cdot \mathrm{W}^{s}_{k}
\]
Transformer_capex
\[
\mathit{Transformer\_capex} = \sum_{\xi \in \Xi,\ m \in \mathcal{M}} \pi_{\xi} \cdot \Sigma_{m} \cdot \mathrm{c}^{\mathrm{cap},\sigma}_{\xi,m} \cdot \mathrm{W}^{\sigma}_{m}
\]
risk_weighted_opex
\[
\mathit{risk\_weighted\_opex} = \sum_{\xi \in \Xi} \pi_{\xi} \cdot \mathit{scenario\_opex}_{\xi}
\]
total_cost
\[
\mathit{total\_cost} = \mathit{Line\_capex} + \mathit{Transformer\_capex} + \mathit{risk\_weighted\_opex}
\]
Load_demand
\[
\mathrm{Load\_demand}_{\xi,t,d} = \begin{cases} \mathrm{sgn}^{\mathrm{load}}_{d} \cdot \mathrm{load}_{\xi,t,d} & \text{if } \mathrm{on}^{\mathrm{load}}_{d} \\ 0 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ d \in \mathcal{D}
\]
Line_s_monitored
\[
\mathit{Line\_s\_monitored}_{\xi,t,k} = \begin{cases} s_{\xi,t,k} & \text{if } \mathrm{on}^{s}_{t,k} \\ 0 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K}
\]
Transformer_s_monitored
\[
\mathit{Transformer\_s\_monitored}_{\xi,t,m} = \begin{cases} \sigma_{\xi,t,m} & \text{if } \mathrm{on}^{\sigma}_{t,m} \\ 0 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M}
\]
Outage_s
\[
\mathit{Outage\_s}_{\xi,t,\kappa} = \begin{cases} \mathit{Line\_s\_monitored}_{\xi,t,\mathrm{Outage\_line}(\kappa)} & \text{if } \mathrm{Outage\_line}(\kappa) \text{ is defined} \\ \mathit{Transformer\_s\_monitored}_{\xi,t,\mathrm{Outage\_transformer}(\kappa)} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ \kappa \in \mathcal{K}^{\mathrm{out}}
\]
Generator_injection
\[
\mathit{Generator\_injection}_{n,\xi,t} = \sum_{g \in \mathcal{G} \,:\, \mathrm{Generator\_bus}(g) = n} \mathrm{sgn}_{g} \cdot p_{\xi,t,g} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Line_injection
\[
\mathit{Line\_injection}_{n,\xi,t} = -\left( \sum_{k \in \mathcal{K} \,:\, \mathrm{Line\_bus0}(k) = n} s_{\xi,t,k} \right) + \sum_{k \in \mathcal{K} \,:\, \mathrm{Line\_bus1}(k) = n} s_{\xi,t,k} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Link_injection
\[
\mathit{Link\_injection}_{n,\xi,t} = -\left( \sum_{l \in \mathcal{L} \,:\, \mathrm{Link\_bus0}(l) = n} f_{\xi,t,l} \right) + \sum_{o \in \mathcal{O} \,:\, \mathrm{Link\_output\_bus}(o) = n} \mathit{Link\_output\_arrival}_{\xi,t,o} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Load_injection
\[
\mathrm{Load\_injection}_{n,\xi,t} = \sum_{d \in \mathcal{D} \,:\, \mathrm{Load\_bus}(d) = n} \mathrm{Load\_demand}_{\xi,t,d} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Process_injection
\[
\mathit{Process\_injection}_{n,\xi,t} = \sum_{r \in \mathcal{R} \,:\, \mathrm{Process\_output\_bus}(r) = n} \mathit{Process\_output\_arrival}_{\xi,t,r} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Transformer_injection
\[
\mathit{Transformer\_injection}_{n,\xi,t} = -\left( \sum_{m \in \mathcal{M} \,:\, \mathrm{Transformer\_bus0}(m) = n} \sigma_{\xi,t,m} \right) + \sum_{m \in \mathcal{M} \,:\, \mathrm{Transformer\_bus1}(m) = n} \sigma_{\xi,t,m} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Bus_injection
\[
\mathit{Bus\_injection}_{\xi,t,n} = \mathit{Generator\_injection}_{n,\xi,t} + \mathit{Line\_injection}_{n,\xi,t} + \mathit{Link\_injection}_{n,\xi,t} + \mathrm{Load\_injection}_{n,\xi,t} + \mathit{Process\_injection}_{n,\xi,t} + \mathit{Transformer\_injection}_{n,\xi,t} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Line_angle_sum
\[
\mathit{Line\_angle\_sum}_{c,\xi,t} = \sum_{k \in \mathcal{K}} s_{\xi,t,k} \cdot \mathrm{x}_{\mathrm{snapshot\_period}(t),k,c} \qquad \forall\, c \in \mathcal{C},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Transformer_angle_sum
\[
\mathit{Transformer\_angle\_sum}_{c,\xi,t} = \sum_{m \in \mathcal{M}} \sigma_{\xi,t,m} \cdot \mathrm{x}^{\sigma}_{\mathrm{snapshot\_period}(t),m,c} \qquad \forall\, c \in \mathcal{C},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Cycle_angle_sum
\[
\mathit{Cycle\_angle\_sum}_{\xi,t,c} = \mathit{Line\_angle\_sum}_{c,\xi,t} + \mathit{Transformer\_angle\_sum}_{c,\xi,t} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ c \in \mathcal{C}
\]
Objective¶
\[
\min \mathit{total\_cost}
\]
Subject to¶
Generator_fix_p_lower
\[
p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_fix_p_upper
\[
p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Link_fix_p_lower
\[
f_{\xi,t,l} \ge \underline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_fix_p_upper
\[
f_{\xi,t,l} \le \overline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Line_fix_s_lower
\[
s_{\xi,t,k} \ge -\overline{\mathrm{s}}_{\xi,t,k} \cdot \mathrm{s}^{\mathrm{nom}}_{\xi,k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K} \,:\, \neg \mathrm{ext}^{s}_{k} \wedge \mathrm{on}^{s}_{t,k}
\]
Line_fix_s_upper
\[
s_{\xi,t,k} \le \overline{\mathrm{s}}_{\xi,t,k} \cdot \mathrm{s}^{\mathrm{nom}}_{\xi,k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K} \,:\, \neg \mathrm{ext}^{s}_{k} \wedge \mathrm{on}^{s}_{t,k}
\]
Line_ext_s_lower
\[
s_{\xi,t,k} \ge -\overline{\mathrm{s}}_{\xi,t,k} \cdot S_{k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{on}^{s}_{t,k}
\]
Line_ext_s_upper
\[
s_{\xi,t,k} \le \overline{\mathrm{s}}_{\xi,t,k} \cdot S_{k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{on}^{s}_{t,k}
\]
Line_ext_s_nom_lower
\[
S_{k} \ge \underline{\mathrm{s}}^{\mathrm{nom}}_{\xi,k} \qquad \forall\, \xi \in \Xi,\ k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{act}^{s}_{k}
\]
Line_ext_s_nom_upper
\[
S_{k} \le \overline{\mathrm{s}}^{\mathrm{nom}}_{\xi,k} \qquad \forall\, \xi \in \Xi,\ k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{act}^{s}_{k} \wedge \overline{\mathrm{s}}^{\mathrm{nom}}_{\xi,k} \text{ is defined}
\]
Transformer_fix_s_lower
\[
\sigma_{\xi,t,m} \ge -\overline{\sigma}_{\xi,t,m} \cdot \sigma^{\mathrm{nom}}_{\xi,m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M} \,:\, \neg \mathrm{ext}^{\sigma}_{m} \wedge \mathrm{on}^{\sigma}_{t,m}
\]
Transformer_fix_s_upper
\[
\sigma_{\xi,t,m} \le \overline{\sigma}_{\xi,t,m} \cdot \sigma^{\mathrm{nom}}_{\xi,m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M} \,:\, \neg \mathrm{ext}^{\sigma}_{m} \wedge \mathrm{on}^{\sigma}_{t,m}
\]
Transformer_ext_s_lower
\[
\sigma_{\xi,t,m} \ge -\overline{\sigma}_{\xi,t,m} \cdot \Sigma_{m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M} \,:\, \mathrm{ext}^{\sigma}_{m} \wedge \mathrm{on}^{\sigma}_{t,m}
\]
Transformer_ext_s_upper
\[
\sigma_{\xi,t,m} \le \overline{\sigma}_{\xi,t,m} \cdot \Sigma_{m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M} \,:\, \mathrm{ext}^{\sigma}_{m} \wedge \mathrm{on}^{\sigma}_{t,m}
\]
Transformer_ext_s_nom_lower
\[
\Sigma_{m} \ge \underline{\sigma}^{\mathrm{nom}}_{\xi,m} \qquad \forall\, \xi \in \Xi,\ m \in \mathcal{M} \,:\, \mathrm{ext}^{\sigma}_{m} \wedge \mathrm{act}^{\sigma}_{m}
\]
Transformer_ext_s_nom_upper
\[
\Sigma_{m} \le \overline{\sigma}^{\mathrm{nom}}_{\xi,m} \qquad \forall\, \xi \in \Xi,\ m \in \mathcal{M} \,:\, \mathrm{ext}^{\sigma}_{m} \wedge \mathrm{act}^{\sigma}_{m} \wedge \overline{\sigma}^{\mathrm{nom}}_{\xi,m} \text{ is defined}
\]
Line_fix_s_lower_security
\[
\mathit{Line\_s\_monitored}_{\xi,t,k} + \beta_{\mathrm{snapshot\_period}(t),k,\kappa} \cdot \mathit{Outage\_s}_{\xi,t,\kappa} \ge -\overline{\mathrm{s}}_{\xi,t,k} \cdot \mathrm{s}^{\mathrm{nom}}_{\xi,k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K},\ \kappa \in \mathcal{K}^{\mathrm{out}} \,:\, \neg \mathrm{ext}^{s}_{k} \wedge \beta_{\mathrm{snapshot\_period}(t),k,\kappa} \text{ is defined}
\]
Line_fix_s_upper_security
\[
\mathit{Line\_s\_monitored}_{\xi,t,k} + \beta_{\mathrm{snapshot\_period}(t),k,\kappa} \cdot \mathit{Outage\_s}_{\xi,t,\kappa} \le \overline{\mathrm{s}}_{\xi,t,k} \cdot \mathrm{s}^{\mathrm{nom}}_{\xi,k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K},\ \kappa \in \mathcal{K}^{\mathrm{out}} \,:\, \neg \mathrm{ext}^{s}_{k} \wedge \beta_{\mathrm{snapshot\_period}(t),k,\kappa} \text{ is defined}
\]
Line_ext_s_lower_security
\[
\mathit{Line\_s\_monitored}_{\xi,t,k} + \beta_{\mathrm{snapshot\_period}(t),k,\kappa} \cdot \mathit{Outage\_s}_{\xi,t,\kappa} \ge -\overline{\mathrm{s}}_{\xi,t,k} \cdot S_{k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K},\ \kappa \in \mathcal{K}^{\mathrm{out}} \,:\, \mathrm{ext}^{s}_{k} \wedge \beta_{\mathrm{snapshot\_period}(t),k,\kappa} \text{ is defined}
\]
Line_ext_s_upper_security
\[
\mathit{Line\_s\_monitored}_{\xi,t,k} + \beta_{\mathrm{snapshot\_period}(t),k,\kappa} \cdot \mathit{Outage\_s}_{\xi,t,\kappa} \le \overline{\mathrm{s}}_{\xi,t,k} \cdot S_{k} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ k \in \mathcal{K},\ \kappa \in \mathcal{K}^{\mathrm{out}} \,:\, \mathrm{ext}^{s}_{k} \wedge \beta_{\mathrm{snapshot\_period}(t),k,\kappa} \text{ is defined}
\]
Transformer_fix_s_lower_security
\[
\mathit{Transformer\_s\_monitored}_{\xi,t,m} + \beta^{\sigma}_{\mathrm{snapshot\_period}(t),m,\kappa} \cdot \mathit{Outage\_s}_{\xi,t,\kappa} \ge -\overline{\sigma}_{\xi,t,m} \cdot \sigma^{\mathrm{nom}}_{\xi,m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M},\ \kappa \in \mathcal{K}^{\mathrm{out}} \,:\, \neg \mathrm{ext}^{\sigma}_{m} \wedge \beta^{\sigma}_{\mathrm{snapshot\_period}(t),m,\kappa} \text{ is defined}
\]
Transformer_fix_s_upper_security
\[
\mathit{Transformer\_s\_monitored}_{\xi,t,m} + \beta^{\sigma}_{\mathrm{snapshot\_period}(t),m,\kappa} \cdot \mathit{Outage\_s}_{\xi,t,\kappa} \le \overline{\sigma}_{\xi,t,m} \cdot \sigma^{\mathrm{nom}}_{\xi,m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M},\ \kappa \in \mathcal{K}^{\mathrm{out}} \,:\, \neg \mathrm{ext}^{\sigma}_{m} \wedge \beta^{\sigma}_{\mathrm{snapshot\_period}(t),m,\kappa} \text{ is defined}
\]
Transformer_ext_s_lower_security
\[
\mathit{Transformer\_s\_monitored}_{\xi,t,m} + \beta^{\sigma}_{\mathrm{snapshot\_period}(t),m,\kappa} \cdot \mathit{Outage\_s}_{\xi,t,\kappa} \ge -\overline{\sigma}_{\xi,t,m} \cdot \Sigma_{m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M},\ \kappa \in \mathcal{K}^{\mathrm{out}} \,:\, \mathrm{ext}^{\sigma}_{m} \wedge \beta^{\sigma}_{\mathrm{snapshot\_period}(t),m,\kappa} \text{ is defined}
\]
Transformer_ext_s_upper_security
\[
\mathit{Transformer\_s\_monitored}_{\xi,t,m} + \beta^{\sigma}_{\mathrm{snapshot\_period}(t),m,\kappa} \cdot \mathit{Outage\_s}_{\xi,t,\kappa} \le \overline{\sigma}_{\xi,t,m} \cdot \Sigma_{m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M},\ \kappa \in \mathcal{K}^{\mathrm{out}} \,:\, \mathrm{ext}^{\sigma}_{m} \wedge \beta^{\sigma}_{\mathrm{snapshot\_period}(t),m,\kappa} \text{ is defined}
\]
Kirchhoff_Voltage_Law
\[
\mathit{Cycle\_angle\_sum}_{\xi,t,c} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ c \in \mathcal{C}
\]
Bus_nodal_balance
\[
\mathit{Bus\_injection}_{\xi,t,n} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
The spec, differential/pypsa/rungs/rung_30_security_constrained.yaml — the file projected onto what this rung builds:
description: A plain `n.optimize()`, and its multi-period and stochastic classes, in one file. Every second-stage
quantity spans a `scenario` (a future dispatch is chosen in) and every asset stands in the investment
`period`s its build year and lifetime span. A parameter spans `scenario` exactly when PyPSA reads it
per scenario. Capacity is chosen once, before the future is known, and paid once per active period at
its cost in expectation over the scenarios; operation is the expectation over the scenarios' weights,
with a share priced at the tail through the CVaR rows, which stand only where a risk preference is set.
A plain run feeds one scenario, one period, all-active masks and unit weights, and the model collapses
to the standard one. A security-constrained run copies each branch flow limit once per outage in an
`outage` set that a plain run leaves empty. Which snapshots an asset is active in, a scenario's weight,
and the outage factors are data prep.
dimensions:
scenario: {description: 'the futures dispatch is chosen in, each with a weight'}
snapshot: {description: dispatch periods, dtype: datetime, ordered: true}
bus: {description: network nodes}
generator: {description: 'generating units, each on one bus'}
link: {description: 'controllable connections, each from one bus to the buses it delivers to'}
link_output: {description: 'a link''s output ports, one label per port a link declares — PyPSA''s `bus1`,
`bus2`, … columns read long, so a link of any number of output ports is one term in the balance,
data prep'}
process: {description: 'generalized multi-port converters, each with an internal power that every port
draws or delivers at its own rate'}
process_output: {description: 'a process''s ports, one label per port a process declares — PyPSA''s
`bus0`, `bus1`, … each carry a signed `rate`, so a process of any number of ports is one term in
the balance, data prep'}
load: {description: 'demands, each on one bus'}
line: {description: 'passive branches, each between two buses, their flow set by impedance'}
transformer: {description: 'passive branches between two buses, their flow set by impedance and tap
ratio, with a phase shift fixed or optimised'}
cycle: {description: 'independent cycles of the passive network graph — the cycle basis, data prep.
Each period has its own basis, of the branches that stand in it; a label is a position in that period''s
basis, so one label names a different cycle in another period'}
outage: {description: 'the passive branches a security-constrained run takes out one at a time — PyPSA''s
`branch_outages`, each a line or a transformer; none on a plain run'}
period: {description: investment periods — PyPSA's `investment_periods`, dtype: int, ordered: true}
relations:
snapshot_period: {description: the investment period a snapshot falls in, key: snapshot, values: period}
Generator_bus: {description: the bus a generator sits on, key: generator, values: bus}
Link_bus0: {description: the bus a link leaves, key: link, values: bus}
Link_output_link: {description: the link an output port belongs to, key: link_output, values: link}
Link_output_bus: {description: 'the bus an output port delivers to — PyPSA''s `bus1`, `bus2`, … columns.
A link of three output ports is three labels here rather than a third relation, so the file states
any number of them', key: link_output, values: bus}
Process_output_process: {description: the process a port belongs to, key: process_output, values: process}
Process_output_bus: {description: 'the bus a port draws from or delivers to — PyPSA''s `bus0`, `bus1`,
… columns. A process of three ports is three labels here rather than a third relation, so the file
states any number of them', key: process_output, values: bus}
Load_bus: {description: the bus a load sits on, key: load, values: bus}
Line_bus0: {description: the bus a line's flow is measured at, key: line, values: bus}
Line_bus1: {description: the bus at a line's other end, key: line, values: bus}
Transformer_bus0: {description: the bus a transformer's flow is measured at, key: transformer, values: bus}
Transformer_bus1: {description: the bus at a transformer's other end, key: transformer, values: bus}
Outage_line:
description: the line an outage takes out; an outage of a transformer has no row
key: outage
values: line
missing: absent
Outage_transformer:
description: the transformer an outage takes out; an outage of a line has no row
key: outage
values: transformer
missing: absent
parameters:
snapshot_weightings_objective:
description: PyPSA's `snapshot_weightings.objective` — hours a snapshot stands for in the cost
dims: [snapshot]
Generator_p_nom:
description: nominal power
dims: [scenario, generator]
Generator_p_nom_extendable:
description: whether the nominal power is a decision
dims: [generator]
dtype: bool
Generator_p_min_pu:
description: least output, per unit of nominal power
dims: [scenario, snapshot, generator]
Generator_p_max_pu:
description: most output, per unit of nominal power — an availability profile
dims: [scenario, snapshot, generator]
Generator_marginal_cost:
description: cost of one unit of output
dims: [scenario, snapshot, generator]
Generator_marginal_cost_quadratic:
description: cost of the square of one unit of output
dims: [scenario, snapshot, generator]
Generator_sign:
description: the sign output enters its bus's balance with — PyPSA's `sign`, `1` unless given, `-1`
for a unit that draws power. PyPSA refuses one that differs by scenario (`constants.py:43`)
dims: [generator]
Generator_committable:
description: whether output is gated by an on/off status decision
dims: [generator]
dtype: bool
Link_p_nom:
description: nominal power
dims: [scenario, link]
Link_p_nom_extendable:
description: whether the nominal power is a decision
dims: [link]
dtype: bool
Link_p_min_pu:
description: least flow, per unit of nominal power — negative for a link that carries both ways
dims: [scenario, snapshot, link]
Link_p_max_pu:
description: most flow, per unit of nominal power
dims: [scenario, snapshot, link]
Link_efficiency:
description: share of the flow that arrives at an output port, PyPSA's `efficiency`, `efficiency2`,
… read long — negative where that port consumes rather than delivers. Read at the snapshot the flow
arrives, so a delayed port delivers at its arrival snapshot's efficiency (`constraints.py:1498`)
dims: [scenario, snapshot, link_output]
Link_output_delay:
description: snapshots a port's delivery lags its link's flow — PyPSA's `delay`, `delay2`, … read
long, divided by the `snapshot_weightings.generators` value and rounded up in data prep; zero for
a port that delivers at once. A port has the same delay in every scenario, because PyPSA refuses
a delay that differs by scenario (`constants.py:52`)
dims: [link_output]
dtype: int
Link_output_cyclic_delay:
description: whether a delayed port's flow wraps from the end of its investment period — PyPSA's `cyclic_delay`,
`cyclic_delay2`, …; where it does not, the flow still in transit at each period's first snapshots
is lost. The flag is the same in every scenario, as the delay is
dims: [link_output]
dtype: bool
Link_marginal_cost:
description: cost of one unit of flow
dims: [scenario, snapshot, link]
Link_marginal_cost_quadratic:
description: cost of the square of one unit of flow
dims: [scenario, snapshot, link]
Link_committable:
description: whether flow is gated by an on/off status decision
dims: [link]
dtype: bool
Process_rate:
description: the energy a port draws or delivers per unit of internal power, PyPSA's `rate0`, `rate1`,
… read long — negative where the port withdraws, positive where it injects; a link is a process
whose `bus0` rate is minus one and whose output rates are its efficiencies. Read at the snapshot
the transfer arrives, so a delayed port transfers at its arrival snapshot's rate (`constraints.py:1498`)
dims: [scenario, snapshot, process_output]
Process_output_delay:
description: snapshots a port's transfer lags its process's internal power — PyPSA's `delay0`, `delay1`,
… read long, divided by the `snapshot_weightings.generators` value and rounded up in data prep;
zero for a port that transfers at once. A port has the same delay in every scenario, as a link's
port has
dims: [process_output]
dtype: int
Process_output_cyclic_delay:
description: whether a delayed port's transfer wraps from the end of its investment period — PyPSA's
`cyclic_delay0`, `cyclic_delay1`, …; where it does not, the energy still in transit at each period's
first snapshots is lost. The flag is the same in every scenario, as the delay is
dims: [process_output]
dtype: bool
Load_p_set:
description: demand
dims: [scenario, snapshot, load]
Load_sign:
description: the sign a load's demand enters its bus's balance with — PyPSA's `sign`, `-1` unless
given, `1` for a load that feeds its bus. PyPSA refuses one that differs by scenario (`constants.py:43`)
dims: [load]
Load_active:
description: whether a load stands in the model — PyPSA's `active`. A load has no build year and no
lifetime, so the flag holds in every snapshot. PyPSA refuses one that differs by scenario (`constants.py:51`)
dims: [load]
dtype: bool
scenario_weight:
description: PyPSA's `scenario_weightings.weight` — the probability of a future
dims: [scenario]
period_weight_objective:
description: PyPSA's `investment_period_weightings.objective` — what a period's cost weighs; PyPSA
reads it only under `multi_investment_periods`, so data prep feeds one otherwise, whatever the column
holds (`optimize.py:205-207`, `:264-266`)
dims: [period]
Generator_active:
description: whether a generator stands in a snapshot's period — PyPSA's `active`, from build year
and lifetime, data prep
dims: [snapshot, generator]
dtype: bool
Link_active:
description: whether a link stands in a snapshot's period — PyPSA's `active`, data prep
dims: [snapshot, link]
dtype: bool
Line_active:
description: whether a line stands in a snapshot's period — PyPSA's `active`, data prep
dims: [snapshot, line]
dtype: bool
Process_active:
description: whether a process stands in a snapshot's period — PyPSA's `active`, data prep
dims: [snapshot, process]
dtype: bool
Transformer_active:
description: whether a transformer stands in a snapshot's period — PyPSA's `active`, data prep
dims: [snapshot, transformer]
dtype: bool
Line_active_asset:
description: whether a line is one of PyPSA's `active_assets` — its `active` flag alone, without build
year and lifetime, so a unit that stands in no snapshot of the horizon still counts. PyPSA refuses
one that differs by scenario (`constants.py:51`)
dims: [line]
dtype: bool
Transformer_active_asset:
description: whether a transformer is one of PyPSA's `active_assets` — its `active` flag alone, without
build year and lifetime, so a unit that stands in no snapshot of the horizon still counts. PyPSA
refuses one that differs by scenario (`constants.py:51`)
dims: [transformer]
dtype: bool
Line_capital_weight:
description: the sum of period weights a line stands in — PyPSA's `active * period_weighting`, summed,
data prep
dims: [line]
Transformer_capital_weight:
description: the sum of period weights a transformer stands in — PyPSA's `active * period_weighting`,
summed, data prep
dims: [transformer]
Line_s_nom:
description: nominal apparent power
dims: [scenario, line]
Line_s_nom_extendable:
description: whether the nominal apparent power is a decision
dims: [line]
dtype: bool
Line_s_max_pu:
description: most flow either way, per unit of nominal apparent power
dims: [scenario, snapshot, line]
Line_s_nom_min:
description: least nominal apparent power an extendable line may be built at
dims: [scenario, line]
Line_s_nom_max:
description: most nominal apparent power an extendable line may be built at
dims: [scenario, line]
Line_capital_cost:
description: 'cost of one unit of nominal apparent power for the modelled horizon — PyPSA''s `periodized_cost`:
`overnight_cost` as an annuity over `lifetime` at `discount_rate`, times `nyears`, where it is given,
and `capital_cost` where it is not, plus `fom_cost` (`components.py:1126-1147`, `costs.py:102-203`),
data prep'
dims: [scenario, line]
Line_cycle_weight:
description: the line's series impedance, signed by its orientation in the cycle — the cycle basis,
data prep; a line in no cycle has no row. PyPSA builds the cycle basis from the first scenario only
(`networks.py:1430-1437`)
dims: [period, line, cycle]
missing: neutral
Line_BODF:
description: the share of an outaged branch's flow a line takes on when that branch goes out — PyPSA's
`BODF`, from the PTDF of the sub-network the period's active branches form, data prep; a row only
where the line and the outage are active in the period and share a sub-network, -1 at the outaged
line itself. The factors are the same in every scenario, because PyPSA takes those of the first
scenario (`abstract.py:534`)
dims: [period, line, outage]
missing: neutral
Transformer_s_nom:
description: nominal apparent power
dims: [scenario, transformer]
Transformer_s_nom_extendable:
description: whether the nominal apparent power is a decision
dims: [transformer]
dtype: bool
Transformer_s_max_pu:
description: most flow either way, per unit of nominal apparent power
dims: [scenario, snapshot, transformer]
Transformer_s_nom_min:
description: least nominal apparent power an extendable transformer may be built at
dims: [scenario, transformer]
Transformer_s_nom_max:
description: most nominal apparent power an extendable transformer may be built at
dims: [scenario, transformer]
Transformer_capital_cost:
description: 'cost of one unit of nominal apparent power for the modelled horizon — PyPSA''s `periodized_cost`:
`overnight_cost` as an annuity over `lifetime` at `discount_rate`, times `nyears`, where it is given,
and `capital_cost` where it is not, plus `fom_cost` (`components.py:1126-1147`, `costs.py:102-203`),
data prep'
dims: [scenario, transformer]
Transformer_cycle_weight:
description: the transformer's effective series reactance, `x` times its tap ratio, signed by its
orientation in the cycle — PyPSA's `x_pu_eff`, the cycle basis, data prep; a transformer in no cycle
has no row. From the first scenario only, as a line's
dims: [period, transformer, cycle]
missing: neutral
Transformer_BODF:
description: the share of an outaged branch's flow a transformer takes on when that branch goes out,
as a line's; a row only where the transformer and the outage are active in the period and share
a sub-network
dims: [period, transformer, outage]
missing: neutral
variables:
Generator_p:
description: '`Generator-p` — output of a generator in a snapshot'
dims: [scenario, snapshot, generator]
where: Generator_active
Link_p:
description: '`Link-p` — PyPSA''s `p0`, the flow measured at the `Link_bus0` end: a positive value
withdraws there and injects at every bus the link''s output ports deliver to'
dims: [scenario, snapshot, link]
where: Link_active
Line_s:
description: '`Line-s` — PyPSA''s `p0`, the flow measured at the `Line_bus0` end: a positive value
withdraws there and injects at `Line_bus1`, lossless'
dims: [scenario, snapshot, line]
where: Line_active
Transformer_s:
description: '`Transformer-s` — PyPSA''s `p0`, the flow measured at the `Transformer_bus0` end: a
positive value withdraws there and injects at `Transformer_bus1`, lossless'
dims: [scenario, snapshot, transformer]
where: Transformer_active
Line_s_nom_ext:
description: '`Line-s_nom` — nominal apparent power where it is a decision, for a line that is active;
the parameter of the same PyPSA name carries the fixed regime'
dims: [line]
where: Line_s_nom_extendable AND Line_active_asset
Transformer_s_nom_ext:
description: '`Transformer-s_nom` — nominal apparent power where it is a decision, for a transformer
that is active; the parameter of the same PyPSA name carries the fixed regime'
dims: [transformer]
where: Transformer_s_nom_extendable AND Transformer_active_asset
Process_p:
description: '`Process-p` — PyPSA''s internal power `p`: a positive value drives every port at its
own rate, withdrawing where the rate is negative and injecting where it is positive'
dims: [scenario, snapshot, process]
where: Process_active
constraints:
Generator_fix_p_lower:
description: '`Generator-fix-p-lower` — a fixed generator outputs at least its minimum'
dims: [scenario, snapshot, generator]
where: not Generator_p_nom_extendable AND not Generator_committable AND Generator_active
expression: Generator_p >= Generator_p_min_pu * Generator_p_nom
Generator_fix_p_upper:
description: '`Generator-fix-p-upper` — a fixed generator outputs at most what is available'
dims: [scenario, snapshot, generator]
where: not Generator_p_nom_extendable AND not Generator_committable AND Generator_active
expression: Generator_p <= Generator_p_max_pu * Generator_p_nom
Link_fix_p_lower:
description: '`Link-fix-p-lower` — a fixed link carries at least its minimum, negative for the other
way'
dims: [scenario, snapshot, link]
where: not Link_p_nom_extendable AND not Link_committable AND Link_active
expression: Link_p >= Link_p_min_pu * Link_p_nom
Link_fix_p_upper:
description: '`Link-fix-p-upper` — a fixed link carries at most its nominal power'
dims: [scenario, snapshot, link]
where: not Link_p_nom_extendable AND not Link_committable AND Link_active
expression: Link_p <= Link_p_max_pu * Link_p_nom
Line_fix_s_lower:
description: '`Line-fix-s-lower` — a fixed line carries at least the negative of its rating, the loss
counted against it'
dims: [scenario, snapshot, line]
where: not Line_s_nom_extendable AND Line_active
expression: Line_s >= (-Line_s_max_pu) * Line_s_nom
Line_fix_s_upper:
description: '`Line-fix-s-upper` — a fixed line carries at most its rating, the loss included'
dims: [scenario, snapshot, line]
where: not Line_s_nom_extendable AND Line_active
expression: Line_s <= Line_s_max_pu * Line_s_nom
Line_ext_s_lower:
description: '`Line-ext-s-lower` — an extendable line carries at least the negative of its rating
of the chosen build, the loss counted against it'
dims: [scenario, snapshot, line]
where: Line_s_nom_extendable AND Line_active
expression: Line_s >= (-Line_s_max_pu) * Line_s_nom_ext
Line_ext_s_upper:
description: '`Line-ext-s-upper` — an extendable line carries at most its rating of the chosen build,
the loss included'
dims: [scenario, snapshot, line]
where: Line_s_nom_extendable AND Line_active
expression: Line_s <= Line_s_max_pu * Line_s_nom_ext
Line_ext_s_nom_lower:
description: '`Line-ext-s_nom-lower` — the chosen build is at least its floor in every scenario'
dims: [scenario, line]
where: Line_s_nom_extendable AND Line_active_asset
expression: Line_s_nom_ext >= Line_s_nom_min
Line_ext_s_nom_upper:
description: '`Line-ext-s_nom-upper` — the chosen build is at most its cap in every scenario; a cap
of infinity is no row'
dims: [scenario, line]
where: Line_s_nom_extendable AND Line_active_asset AND Line_s_nom_max
expression: Line_s_nom_ext <= Line_s_nom_max
Transformer_fix_s_lower:
description: '`Transformer-fix-s-lower` — a fixed transformer carries at least the negative of its
rating, the loss counted against it'
dims: [scenario, snapshot, transformer]
where: not Transformer_s_nom_extendable AND Transformer_active
expression: Transformer_s >= (-Transformer_s_max_pu) * Transformer_s_nom
Transformer_fix_s_upper:
description: '`Transformer-fix-s-upper` — a fixed transformer carries at most its rating, the loss
included'
dims: [scenario, snapshot, transformer]
where: not Transformer_s_nom_extendable AND Transformer_active
expression: Transformer_s <= Transformer_s_max_pu * Transformer_s_nom
Transformer_ext_s_lower:
description: '`Transformer-ext-s-lower` — an extendable transformer carries at least the negative
of its rating of the chosen build, the loss counted against it'
dims: [scenario, snapshot, transformer]
where: Transformer_s_nom_extendable AND Transformer_active
expression: Transformer_s >= (-Transformer_s_max_pu) * Transformer_s_nom_ext
Transformer_ext_s_upper:
description: '`Transformer-ext-s-upper` — an extendable transformer carries at most its rating of
the chosen build, the loss included'
dims: [scenario, snapshot, transformer]
where: Transformer_s_nom_extendable AND Transformer_active
expression: Transformer_s <= Transformer_s_max_pu * Transformer_s_nom_ext
Transformer_ext_s_nom_lower:
description: '`Transformer-ext-s_nom-lower` — the chosen build is at least its floor in every scenario'
dims: [scenario, transformer]
where: Transformer_s_nom_extendable AND Transformer_active_asset
expression: Transformer_s_nom_ext >= Transformer_s_nom_min
Transformer_ext_s_nom_upper:
description: '`Transformer-ext-s_nom-upper` — the chosen build is at most its cap in every scenario;
a cap of infinity is no row'
dims: [scenario, transformer]
where: Transformer_s_nom_extendable AND Transformer_active_asset AND Transformer_s_nom_max
expression: Transformer_s_nom_ext <= Transformer_s_nom_max
Line_fix_s_lower_security:
description: '`Line-fix-s-lower-security-for-{c}-outage-in-sub-network-{n}` — after any one outage,
a fixed line carries at least the negative of its rating: its flow takes on its share of the outaged
branch''s flow. PyPSA names one row per outaged component `c` and sub-network `n`, and under `multi_investment_periods`
it appends `-period-{p}` for each period `p`. This entry states them all over the outage dimension'
dims: [scenario, snapshot, line, outage]
where: not Line_s_nom_extendable AND at(Line_BODF, by=snapshot_period[period])
expression: Line_s_monitored + at(Line_BODF, by=snapshot_period[period]) * Outage_s >= -Line_s_max_pu
* Line_s_nom
Line_fix_s_upper_security:
description: '`Line-fix-s-upper-security-for-{c}-outage-in-sub-network-{n}` — after any one outage,
a fixed line carries at most its rating: its flow takes on its share of the outaged branch''s flow.
PyPSA names one row per outaged component `c` and sub-network `n`, and under `multi_investment_periods`
it appends `-period-{p}` for each period `p`. This entry states them all over the outage dimension'
dims: [scenario, snapshot, line, outage]
where: not Line_s_nom_extendable AND at(Line_BODF, by=snapshot_period[period])
expression: Line_s_monitored + at(Line_BODF, by=snapshot_period[period]) * Outage_s <= Line_s_max_pu
* Line_s_nom
Line_ext_s_lower_security:
description: '`Line-ext-s-lower-security-for-{c}-outage-in-sub-network-{n}` — after any one outage,
an extendable line carries at least the negative of its rating of the chosen build: its flow takes
on its share of the outaged branch''s flow. PyPSA names one row per outaged component `c` and sub-network
`n`, and under `multi_investment_periods` it appends `-period-{p}` for each period `p`. This entry
states them all over the outage dimension'
dims: [scenario, snapshot, line, outage]
where: Line_s_nom_extendable AND at(Line_BODF, by=snapshot_period[period])
expression: Line_s_monitored + at(Line_BODF, by=snapshot_period[period]) * Outage_s >= -Line_s_max_pu
* Line_s_nom_ext
Line_ext_s_upper_security:
description: '`Line-ext-s-upper-security-for-{c}-outage-in-sub-network-{n}` — after any one outage,
an extendable line carries at most its rating of the chosen build: its flow takes on its share of
the outaged branch''s flow. PyPSA names one row per outaged component `c` and sub-network `n`, and
under `multi_investment_periods` it appends `-period-{p}` for each period `p`. This entry states
them all over the outage dimension'
dims: [scenario, snapshot, line, outage]
where: Line_s_nom_extendable AND at(Line_BODF, by=snapshot_period[period])
expression: Line_s_monitored + at(Line_BODF, by=snapshot_period[period]) * Outage_s <= Line_s_max_pu
* Line_s_nom_ext
Transformer_fix_s_lower_security:
description: '`Transformer-fix-s-lower-security-for-{c}-outage-in-sub-network-{n}` — after any one
outage, a fixed transformer carries at least the negative of its rating: its flow takes on its share
of the outaged branch''s flow. PyPSA names one row per outaged component `c` and sub-network `n`,
and under `multi_investment_periods` it appends `-period-{p}` for each period `p`. This entry states
them all over the outage dimension'
dims: [scenario, snapshot, transformer, outage]
where: not Transformer_s_nom_extendable AND at(Transformer_BODF, by=snapshot_period[period])
expression: Transformer_s_monitored + at(Transformer_BODF, by=snapshot_period[period]) * Outage_s
>= -Transformer_s_max_pu * Transformer_s_nom
Transformer_fix_s_upper_security:
description: '`Transformer-fix-s-upper-security-for-{c}-outage-in-sub-network-{n}` — after any one
outage, a fixed transformer carries at most its rating: its flow takes on its share of the outaged
branch''s flow. PyPSA names one row per outaged component `c` and sub-network `n`, and under `multi_investment_periods`
it appends `-period-{p}` for each period `p`. This entry states them all over the outage dimension'
dims: [scenario, snapshot, transformer, outage]
where: not Transformer_s_nom_extendable AND at(Transformer_BODF, by=snapshot_period[period])
expression: Transformer_s_monitored + at(Transformer_BODF, by=snapshot_period[period]) * Outage_s
<= Transformer_s_max_pu * Transformer_s_nom
Transformer_ext_s_lower_security:
description: '`Transformer-ext-s-lower-security-for-{c}-outage-in-sub-network-{n}` — after any one
outage, an extendable transformer carries at least the negative of its rating of the chosen build:
its flow takes on its share of the outaged branch''s flow. PyPSA names one row per outaged component
`c` and sub-network `n`, and under `multi_investment_periods` it appends `-period-{p}` for each
period `p`. This entry states them all over the outage dimension'
dims: [scenario, snapshot, transformer, outage]
where: Transformer_s_nom_extendable AND at(Transformer_BODF, by=snapshot_period[period])
expression: Transformer_s_monitored + at(Transformer_BODF, by=snapshot_period[period]) * Outage_s
>= -Transformer_s_max_pu * Transformer_s_nom_ext
Transformer_ext_s_upper_security:
description: '`Transformer-ext-s-upper-security-for-{c}-outage-in-sub-network-{n}` — after any one
outage, an extendable transformer carries at most its rating of the chosen build: its flow takes
on its share of the outaged branch''s flow. PyPSA names one row per outaged component `c` and sub-network
`n`, and under `multi_investment_periods` it appends `-period-{p}` for each period `p`. This entry
states them all over the outage dimension'
dims: [scenario, snapshot, transformer, outage]
where: Transformer_s_nom_extendable AND at(Transformer_BODF, by=snapshot_period[period])
expression: Transformer_s_monitored + at(Transformer_BODF, by=snapshot_period[period]) * Outage_s
<= Transformer_s_max_pu * Transformer_s_nom_ext
Kirchhoff_Voltage_Law:
description: '`Kirchhoff-Voltage-Law` — around every independent cycle the impedance-weighted flows
sum to nothing, which is what makes the linear power flow physical rather than transport. A transformer''s
flow weighs its effective reactance, and its phase shift enters the cycle sum too: a constant where
the shift is fixed, or the shift decision times its cycle weight where the shift is a phase-shifting
transformer''s to choose. A snapshot reads the cycles of its own period, of the branches that stand
in it (`constraints.py:1622-1638`); a cycle label that period''s basis does not reach has no row'
dims: [scenario, snapshot, cycle]
expression: Cycle_angle_sum == 0
Bus_nodal_balance:
description: '`Bus-nodal_balance` — what is generated at a bus, storage dispatch and stores included,
less what the links take away, plus what arrives over them after losses and any delay at every port
they deliver to, each process port drawing or delivering at its own rate and each passive branch
carrying its flow, meets the load there, less half of every incident line''s and transformer''s
loss — PyPSA dissipates a branch''s loss half at either end. Each generator, storage unit, store
and load term enters with its component''s `sign` (`constraints.py:1404-1405`, `:1514`), and an
inactive load not at all. A bus nothing is attached to has no row; PyPSA refuses one that carries
load, and this file does not yet.'
dims: [scenario, snapshot, bus]
expression: Bus_injection == 0
expressions:
Link_output_arrival:
description: what a link delivers to an output port at a snapshot — its flow delayed by the port's
`delay` within its investment period, times the port's efficiency at the snapshot the flow arrives;
where the port is `cyclic_delay` the delayed flow wraps from the period's end, and where it is not
the flow still in transit at the period's first snapshots is lost. A port that does not delay (`delay`
zero) delivers its flow unshifted, cyclic or not
dims: [scenario, snapshot, link_output]
cases:
wrapping: {when: Link_output_cyclic_delay, expression: 'shift(at(Link_p, by=Link_output_link[link]),
along=snapshot, offset=Link_output_delay, edge=''wrap'', within=snapshot_period[period]) * Link_efficiency'}
otherwise: shift(at(Link_p, by=Link_output_link[link]), along=snapshot, offset=Link_output_delay,
edge=0, within=snapshot_period[period]) * Link_efficiency
Process_output_arrival:
description: what a process transfers at a port at a snapshot — its internal power delayed by the
port's `delay` within its investment period, times the port's rate at the snapshot the transfer
arrives; where the port is `cyclic_delay` the delayed transfer wraps from the period's end, and
where it is not the energy still in transit at the period's first snapshots is lost. A port that
does not delay (`delay` zero) transfers at once, cyclic or not
dims: [scenario, snapshot, process_output]
cases:
wrapping: {when: Process_output_cyclic_delay, expression: 'shift(at(Process_p, by=Process_output_process[process]),
along=snapshot, offset=Process_output_delay, edge=''wrap'', within=snapshot_period[period])
* Process_rate'}
otherwise: shift(at(Process_p, by=Process_output_process[process]), along=snapshot, offset=Process_output_delay,
edge=0, within=snapshot_period[period]) * Process_rate
Generator_opex: {expression: 'sum(sum(((Generator_p * Generator_marginal_cost) * snapshot_weightings_objective)
* at(period_weight_objective, by=snapshot_period[period]), over=generator), over=snapshot) + sum(sum((((Generator_p
* Generator_p) * Generator_marginal_cost_quadratic) * snapshot_weightings_objective) * at(period_weight_objective,
by=snapshot_period[period]), over=generator), over=snapshot)'}
Link_opex: {expression: 'sum(sum(((Link_p * Link_marginal_cost) * snapshot_weightings_objective) * at(period_weight_objective,
by=snapshot_period[period]), over=link), over=snapshot) + sum(sum((((Link_p * Link_p) * Link_marginal_cost_quadratic)
* snapshot_weightings_objective) * at(period_weight_objective, by=snapshot_period[period]), over=link),
over=snapshot)'}
scenario_opex:
dims: [scenario]
expression: Generator_opex + Link_opex
description: what a future costs to run — every operating term, weighted by the snapshot's hours and
its period, before the scenario's own weight; a start and a stop cost what they cost, unweighted,
as PyPSA adds them (`optimize.py:415-432`)
Line_capex: {expression: sum(scenario_weight * Line_s_nom_ext * Line_capital_cost * Line_capital_weight)}
Transformer_capex: {expression: sum(scenario_weight * Transformer_s_nom_ext * Transformer_capital_cost
* Transformer_capital_weight)}
risk_weighted_opex: {expression: 'sum(scenario_weight * scenario_opex, over=scenario)'}
total_cost:
dims: []
expression: (Line_capex + Transformer_capex) + risk_weighted_opex
description: what the system costs — capacity once per active period at its expected cost over the
scenarios, operation in expectation over the scenarios, and a share of it at the tail
Load_demand:
description: what a load draws from its bus's balance — its demand times its sign where it is active,
nothing where it is not, since PyPSA drops an inactive load from the balance (`constraints.py:1513-1514`)
dims: [scenario, snapshot, load]
cases:
active: {when: Load_active, expression: Load_sign * Load_p_set}
otherwise: 0
Line_s_monitored:
description: the flow a line's post-contingency rows read — its flow where it stands, nothing where
it does not, since PyPSA builds those rows for every branch of the sub-network in every snapshot
dims: [scenario, snapshot, line]
cases:
standing: {when: Line_active, expression: Line_s}
otherwise: 0
Transformer_s_monitored:
description: the flow a transformer's post-contingency rows read, as a line's
dims: [scenario, snapshot, transformer]
cases:
standing: {when: Transformer_active, expression: Transformer_s}
otherwise: 0
Outage_s:
description: the flow an outage takes off its branch — the outaged line's or transformer's flow before
it goes out
dims: [scenario, snapshot, outage]
cases:
line: {when: Outage_line, expression: 'at(Line_s_monitored, by=Outage_line[line])'}
otherwise: at(Transformer_s_monitored, by=Outage_transformer[transformer])
Generator_injection: {expression: 'sum(Generator_sign * Generator_p, over=generator, by=Generator_bus[bus])'}
Line_injection: {expression: '(-sum(Line_s, over=line, by=Line_bus0[bus])) + sum(Line_s, over=line,
by=Line_bus1[bus])'}
Link_injection: {expression: '-sum(Link_p, over=link, by=Link_bus0[bus]) + sum(Link_output_arrival,
over=link_output, by=Link_output_bus[bus])'}
Load_injection: {expression: 'sum(Load_demand, over=load, by=Load_bus[bus])'}
Process_injection: {expression: 'sum(Process_output_arrival, over=process_output, by=Process_output_bus[bus])'}
Transformer_injection: {expression: '(-sum(Transformer_s, over=transformer, by=Transformer_bus0[bus]))
+ sum(Transformer_s, over=transformer, by=Transformer_bus1[bus])'}
Bus_injection:
dims: [scenario, snapshot, bus]
expression: ((((Generator_injection + Line_injection) + Link_injection) + Load_injection) + Process_injection)
+ Transformer_injection
description: what every component puts into a bus, less what it takes out of it; PyPSA writes each
term into the balance, and a load on its right-hand side
Line_angle_sum: {expression: 'sum(Line_s * at(Line_cycle_weight, by=snapshot_period[period]), over=line)'}
Transformer_angle_sum: {expression: 'sum(Transformer_s * at(Transformer_cycle_weight, by=snapshot_period[period]),
over=transformer)'}
Cycle_angle_sum:
dims: [scenario, snapshot, cycle]
expression: Line_angle_sum + Transformer_angle_sum
description: 'the voltage angle differences around a cycle: every branch flow times its cycle weight,
and every transformer phase shift'
objective: {sense: minimize, expression: total_cost}
The prep — every table the spec declares, from the network — and the solve:
from differential.pypsa.prep import relation, static, varying, weighting
n = build() # the network from the PyPSA tab
sources = {
'snapshot': pl.Series('snapshot', list(timesteps(n)), dtype=pl.Datetime('us')),
'bus': pl.Series('bus', list(names(n.buses.index).astype(str)), dtype=pl.String),
**{
dim: pl.Series(dim, list(names(n.static(component).index).astype(str)), dtype=pl.String)
for component, dim in DIM.items()
},
**scenarios(n),
**periods(n, multi),
**carriers(n, multi),
'Generator_bus': relation(n, 'Generator', 'bus'),
'Link_bus0': relation(n, 'Link', 'bus0'),
'Load_bus': relation(n, 'Load', 'bus'),
'snapshot_weightings_objective': weighting(n, 'objective'),
'Generator_sign': per_component('Generator', first_scenario(n.generators['sign'])),
'Load_p_set': varying(n, 'Load', 'p_set'),
'Load_sign': per_component('Load', first_scenario(loads['sign'])),
'Load_active': per_component('Load', first_scenario(loads['active']), bool),
}
with sps.solve('differential/pypsa/rungs/rung_30_security_constrained.yaml', sources) as solution:
solution.objective # 22113.333333
The network, rung_30_security_constrained.py in the corpus — the spine plus what this rung adds:
# SPDX-FileCopyrightText: mathspec Contributors
#
# SPDX-License-Identifier: MIT
"""Rung 30: security-constrained — a meshed triangle of lines and two transformers, one extendable, that must carry their flow within their rating after any one of three outages."""
from __future__ import annotations
import pandas as pd
import spine
BRANCH_OUTAGES = pd.MultiIndex.from_tuples([('Line', 'ab'), ('Line', 'ca'), ('Transformer', 'ca_t')])
def build():
"""The spine plus this rung's additions, as a ``pypsa.Network``."""
n = spine.build()
n.add('Bus', 'a')
n.add('Bus', 'b')
n.add('Bus', 'c')
n.add('Generator', 'hydro30', bus='a', p_nom=100, marginal_cost=10)
n.add('Generator', 'diesel30', bus='b', p_nom=100, marginal_cost=50)
n.add('Generator', 'peak30', bus='c', p_nom=100, marginal_cost=200)
n.add('Load', 'town30', bus='c', p_set=[40, 60, 80, 50])
n.add('Line', 'ab', bus0='a', bus1='b', x=0.1, s_nom=60)
n.add('Line', 'bc', bus0='b', bus1='c', x=0.1, s_nom=60, s_max_pu=0.9)
n.add('Line', 'ca', bus0='c', bus1='a', x=0.1, s_nom_extendable=True, capital_cost=5, s_nom_max=200)
n.add('Transformer', 'ca_t', bus0='c', bus1='a', x=0.2, s_nom=30)
n.add('Transformer', 'bc_t', bus0='b', bus1='c', x=0.3, s_nom_extendable=True, capital_cost=3, s_nom_max=50)
return n
The data¶
The tables this rung is the first to declare (5), as the prep produced them:
Line_BODF.csv
period,line,outage,value
0,ab,Line-ab,-1.0
0,ab,Line-ca,-0.032258064516
0,ab,Transformer-ca_t,-0.333333333333
0,bc,Line-ab,-1.0
0,bc,Line-ca,-0.032258064516
0,bc,Transformer-ca_t,-0.333333333333
0,ca,Line-ab,-0.0625
0,ca,Line-ca,-1.0
0,ca,Transformer-ca_t,0.666666666667
Outage_line.csv
Outage_transformer.csv
Transformer_BODF.csv
period,transformer,outage,value
0,bc_t,Line-ab,0.0
0,bc_t,Line-ca,0.0
0,bc_t,Transformer-ca_t,0.0
0,ca_t,Line-ab,-0.9375
0,ca_t,Line-ca,0.967741935484
0,ca_t,Transformer-ca_t,-1.0
outage.csv